> I don't 100% get the band limited signal bit. How does band limiting imply that there's only a single possible reconstruction of the digital signal? I can kind of picture the fourier transform meaning that there's only one representation but it's a bit of a leap for me. Can the converter itself not create the signal incorrectly?

Band limiting means that there's a maximum frequency. You can think of it as a limit on how fast the signal can change. There's only one possible way to draw a curve that goes through all the points without changing too quickly.

It's something like graphs of polynomials. There's only one possible line (first-degree polynomial) going through any two points. Three points completely determine a second-degree polynomial (parabola), etc. There are an infinite number of parabolas you can draw through two points, just as there are an infinite number of squiggles you could draw through the digital sample points. But you can't create a different solution without adding higher-order terms to the polynomial/higher frequencies to the signal. The number of points you have is sufficient to completely specify the curve.

Though there's only one theoretical solution, the converter can still "create it incorrectly".